Cardinal Structure Under Determinacy
We assume throughout ZF + AD + DC, and develop the cardinal
structure or “cardinal arithmetic” as far as possible.
We describe:
• The “global” results, those that hold throughout the entire
Wadge hierarchy but are of a more general nature.
• The “local” results which require a more detailed inductive analysis, but which provide a detailed understanding of the cardinal
structure. These results are currently only known to extend
through a comparatively small initial segment of the Wadge
hierarchy.
As an application we present a result which links the cardinal
structure of L(R) to that of V .
1
2
Basic Concepts
AD: Every two player integer game is determined.
For A ⊆ ω ω , we have the game GA:
I x(0)
II
x(1)
x(2)
x(3)
x(4) . . .
x(5) . . .
I wins the run iff x ∈ A, where
x = (x(0), x(1), x(2), . . . ).
A strategy (for an integer game) for I (II) is a function σ from
the sequences s ∈ ω <ω of even (odd) length to ω. We say σ is a
winning strategy for I (II) if for all runs x ∈ ω ω of the game where
I (II) has followed σ, we have x ∈ A (x ∈
/ A).
If σ is a strategy for I (or II), and x = (x(1), x(3), . . . ) ∈ ω ω , let
σ(x) ∈ ω ω be the result of following σ against II’s play of x. Note
that σ[ω ω ] is a Σ11 subset of ω ω .
3
We employ variations of this notation, e.g., describe a game by
saying I plays out x, II plays out y. Here we might use σ(y) to
denote just I’s response x following σ against II plays of y. Meaning
should be clear from context.
Concepts generalize in natural ways to games on sets X other
than ω. If X cannot be wellordered in ZF then we usually use
quaistrategies (which assign a non-empty set σ(s) ⊆ X to s ∈ X <ω
of appropriate parity length).
If Γ is a collection of subsets of ω ω , we write det(Γ) to denote that
GA is determined for all A ∈ Γ.
DC: If R ⊆ X <ω and
∀(x0, . . . xn−1) ∈ R ∃xn (x1, . . . , xn−1, xn) ∈ R,
then ∃~x ∈ X ω ∀n (~x n ∈ R).
Equivalent to: Every illfounded relation has an infinite descending
sequence.
4
L(R)
L(R) is the smallest inner-model (i.e., transitive, proper class
model) containing the reals R (which we often identify with ω ω ).
It is defined through a hierarchy similar to L.
J0(R) = Vω+1
[
Jα(R) =
Jβ (R) for α limit.
β<α
Jα+1(R) = rud(Jα(R) ∪ {Jα(R)})
Jα+1(R) =
S
nS
n
(Jα(R)), where S(X) =
S11
i=1 Fi [X
∪ {X}].
Each S n(Jα(R)) is transitive. Every set in L(R) is ordinal definable
from a real. In fact, there is uniformly in α a Σ1(Jα(R)) map from
ωα<ω × R onto Jα(R).
5
Some facts:
(1) AD contradicts ZFC but is consistent with restricted forms of determinacy, e.g., projective determinacy PD, or the determinacy
of games in L(R).
(2) AD is equivalent to ADX , where X is any countable set with at
least two elements.
(3) ADω1 is inconsistent.
(4) ADR is a (presumably) consistent strengthening of AD. It is
equivalent (Martin-Woodin) to AD+ every set has a scale.
(5) AD ⇒ ADL(R).
(6) DC is independent of even ADR (Solovay), but on the other hand
AD ⇒ DCL(R) (Kechris).
(7) AD implies regularity properties for sets of reals, e.g., every
set of reals has the perfect set property, the Baire property,
is measurable, Ramsey. The determinacy needed is local, e.g.,
Π11-det implies perfect set property for Σ12.
(8) Under AD, successor cardinals need not be regular (but cof(κ+) >
ω).
6
Global Results-First Pass: Separation, Reduction,
Prewellordering
Definition. For A, B ⊆ ω ω , we say that A is Wadge reducible to
B, A ≤w B, if there is a continuous function f : ω ω → ω ω such
that A = f −1(B), i.e., x ∈ A iff f (x) ∈ B.
We say A is Lipschitz reducible to B, A ≤` B if there is a Lipschitz continuous f (i.e., a strategy for II) such that A = f −1(B).
For A, B ⊆ ω ω , we have the basic (Lipschitz) Wadge game G`(A, B):
I plays out x, II plays out y, and II wins the run iff (x ∈ A ↔ y ∈
B).
If II has a winning strategy then A ≤` B. If I has a winning
strategy then B ≤` Ac.
We consider pairs {A, Ac}. We say {A, Ac} ≤ {B, B c} if A ≤ B
or A ≤ B c (≤ means ≤` or ≤w ).
7
Theorem (Martin-Monk). ≤` is a wellordering (and hence also
≤w ).
We say A is Lipschitz selfdual if A ≤` Ac, and likewise for Wadge
selfdual. A theorem of Steel says A is Lipschitz selfdual iff A is
Wadge selfdual. We write {A} in this case.
Definition. If A ⊆ ω ω , then o`(A) denotes the rank of {A, Ac} in
≤`. o(A) = ow (A) denotes the rank in ≤w .
The following results suffice to give a complete picture of the ` and
w degrees.
• If A is non-selfdual, then A ⊕ Ac is selfdual and is the next
`-degree after A.
Here A ⊕ B denotes the join
A ⊕ B = {x : (x(0) is even ∧ x0 ∈ A)
∨ (x(0) is odd ∧ x0 ∈ B)},
where x0(k) = x(k + 1).
8
• If A is selfdual, then the next `-degree after A is selfdual and
consists of A0 = {0ax : x ∈ A}.
• At limit ordinals α of cofinality omega there is a selfdual degree
consisting of the countable join of sets of degrees cofinal in α.
• At limit ordinals of uncountable cofinality there is a non-selfdual
degree.
• The next ω1 `-degrees after a selfdual degree are all w-equivalent.
Picture of the w-degrees
•
•
•
•
•
• ··· •
•
•
• ···
•
cof(α) = ω
•
•
•
cof(α) > ω
• ···
•
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Definition. A pointclass is a collection Γ ⊆ P(ω ω ) closed under
Wadge reduction.
We let o(Γ) = sup{o(A) : A ∈ Γ}.
We say Γ is selfdual if A ∈ Γ ⇒ Ac ∈ Γ. Otherwise Γ is nonselfdual.
If Γ is non-selfdual, then Γ has a universal set. Let A ∈ Γ − Γ̌.
Define U ⊆ ω ω × ω ω by U (x, y) ↔ τx(y) ∈ A.
Here every x ∈ ω ω is viewed as a strategy τx for II by: τx(s) =
x(hsi) where s 7→ hsi is a reasonable bijection between ω <ω and ω.
Fact. From universal sets one can construct (in ZF) good universal
sets, that is, universal sets which admit continuous s-m-n functions,
and hence have the recursion theorem.
10
Definition. Γ has the separation property, sep(Γ), if whenever A,
B ∈ Γ and A ∩ B = ∅, then there is a C ∈ ∆ = Γ ∩ Γ̌ with
A ⊆ C, B ∩ C = ∅.
Theorem (Steel-Van Wesep). For any non-selfdual Γ, exactly
one of sep(Γ), sep(Γ̌) holds.
Definition. A norm ϕ on a set A ⊆ ω ω is a map ϕ : A → On.
We say ϕ is regular if ran(ϕ) ∈ On.
Norms ϕ on sets A can be identified with prewellorderings of
A (connected, reflexive, transitive, binary relations on A). The pwo
induces a wellordering on the equivalence classes
[x] = {y : x y ∧ y x}.
The corresponding norm is ϕ(x) = rank of [x].
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Definition. A Γ-norm ϕ on A ⊆ ω ω is a norm such the relations
x <∗ y ↔ x ∈ A ∧ (y ∈
/ A ∨ (y ∈ A ∧ ϕ(x) < ϕ(y)))
x ≤∗ y ↔ x ∈ A ∧ (y ∈
/ A ∨ (y ∈ A ∧ ϕ(x) ≤ ϕ(y)))
are both in Γ.
Definition. Γ has the prewellordering property, pwo(Γ), if every
A ∈ Γ admits a Γ-norm.
Let ϕ : A → θ be a (regular) Γ-norm on A. for α < θ, let
Aα = {x ∈ A : ϕ(x) = α}. So, Aα = A≤α − A<α (in natural
notation).
If x ∈ A and ϕ(x) = α, then:
A≤α = {y : y ≤∗ x} = {y : ¬(x <∗ y)}.
12
So, A≤α ∈ ∆. Likewise A<α ∈ ∆, and so Aα ∈ ∆. So, pwo(Γ)
give an effective way of writing every A ∈ Γ as an increasing union
of ∆ sets.
The initial segment ≺α of the prewellordering can be computed
as:
x ≺α y ↔ x, y ∈ A≤α ∧ (x ≤∗ y)
↔ x, y ∈ A≤α ∧ ¬(y <∗ x)
So, ≺α∈ ∆ if Γ is closed under ∧, ∨.
Definition. δ(Γ) = the supremum of the lengths of the ∆ prewellorderings.
So, if Γ is closed under ∧, ∨, and ϕ is a Γ-norm then |ϕ| ≤ δ(Γ).
13
ω
Fact. If Γ is closed under ∀ω , ∧, ∨, and ϕ is a Γ norm on a
Γ-complete set, then |ϕ| = δ(Γ).
Levy Classes
Definition. Γ is a Levy class if it is a non-selfdual pointclass closed
ωω
ωω
under either ∃ or ∀ (or both).
Σ1α
ωω
We let
enumerate the Levy classes closed under ∃
ωω
closed under ∀ ).
(Π1α those
Σ10 = open, Σ11 = analytic, etc.
Theorem (Steel). For every Levy class Γ, either pwo(Γ) or
pwo(Γ̌).
Steel’s analysis gives more information.
14
.
Let C ⊆ θ be the c.u.b. set of limit α such that Λα = {A : o(A) <
α} is closed under quantifiers.
For Γ a Levy class, let α be the largest element of C such that
Λα ⊆ Γ.
Then Γ is in the projective hierarchy over Λ = Λα. This hierarchy
can fall into one of several types.
Let Γ0 be the non-selfdual pointclass with o(Γ0) = α and w.l.o.g.
sep(Γ̌). We call Γ0 a Steel pointclass. Steel showed Γ0 is closed
ωω
under ∀ .
15
Types of Projective Hierarchies
• Type 1.) cof(α)
S = ω.
Let Σ0 = ω Λ. Then pwo(Σ0), and by periodicity pwo(Π2n+1),
pwo(Σ2n+2).
• Type 2.) cof(α) > ω and Γ0 is not closed under ∨.
Let Π1 = Γ0. Then pwo(Π2n+1), pwo(Σ2n+2).
ω
• Type 3.) cof(α) > ω, Γ0 is closed under ∨ but not ∃ω .
Same conclusion as in type 2.
• Type 4.) Γ0 is closed under quantifiers.
Then pwo(Γ0). Let
ωω
Σ0 = ∃ (Γ0 ∧ Γ̌0).
Then pwo(Σ0), pwo(Π2n+1), pwo(Σ2n+2).
16
Second Pass: Scales and Suslin Cardinals
A tree on X is a subset of X <ω closed under initial segment.
We identify trees on X × Y with subsets of X <ω × Y <ω .
If T is a tree on X, then
[T ] = {f ∈ X ω : ∀n f n ∈ T }.
If T is a tree on X × Y then
p[T ] = {f ∈ X ω : ∃g ∈ Y ω (f, g) ∈ [T ]}
= {f ∈ X ω : Tf is illfounded},
where Tf = {s ∈ Y <ω : (f lh(s), s) ∈ T }.
17
Definition. A ⊆ ω ω is κ-Suslin if A = p[T ] for some tree T on
ω × κ. S(κ) denotes
Sthe pointclass of κ-Suslin sets. κ is a Suslin
cardinal if S(κ) − λ<κ S(λ) 6= ∅.
note: If makes sense to speak of κ-Suslin subsets of λω for λ ∈ On
as well.
Fact. For any Suslin cardinal κ, S(κ) is closed under countable
ω
unions and intersection, ∃ω and (Kechris) is non-selfdual.
Suslin representations ≈ Scales.
Definition. A semi-scale on A ⊆ ω ω is a sequence of norms
{ϕn}n∈ω such that if xn ∈ A, xn → x, and for each n, ϕn(xm)
is eventually equal to some λn, then x ∈ A.
If in addition we have ϕn(x) ≤ λn for all n, then the {ϕn} is said
to be a scale.
18
Fact. For all cardinals κ, A is κ-Suslin iff A admits a semi-scale
with norms into κ iff A admits a scale with norms into κ.
If ϕ
~ is a semi-scale, the corresponding Suslin representation is given
by the tree
Tϕ~ = {((a0, . . . , an−1), (α0, . . . , αn−1)) :
∃x ∈ A (x n = (a0, . . . , an−1)
∧ ∀i < n ϕi(x) = αi}.
If A = p[T ], let for x ∈ A, ϕn(x) = nth coordinate of the leftmost
branch of Tx.
Then ϕ
~ is a semi-scale on A. Let ψn = hϕ0(x), . . . ϕn−1(x)i =
rank of (ϕ0(x), . . . ϕn−1(x)) in lexicographic ordering on κn. Then
~ is a scale on A. [with a little adjustment to T can make the ψi
ψ
map into κ.]
note: If ϕ
~ is a scale, then ϕ
~ (Tϕ~ ) = ϕ
~ . But, not every tree is the
tree of a scale, so we only have Tϕ~ (T ) ⊆ T .
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Theorem (Steel-Woodin). Assume AD. The Suslin cardinals
are closed below their supremum.
Some pointclass arguments together with an analysis of Martin
for constructing the next Suslin gives a classification of the Suslin
cardinals and scales from AD.
Suppose κ is a limit of Suslin cardinals, and κ is below the supremum of the Suslin cardinals (so κ is a Suslin cardinal). We have the
following cases.
• cof(κ) = ω (type
S I).
Let Σ0 = ω S<κ. Then scale(Σ0) with norms into κ, and
.
scale(Π2n+1), scale(Σ2n+2) with norms into δ2n+1 = δ(Π2n+1).
δ2n+1 = (λ2n+1)+, where cof(λ2n+1) = ω (λ1 = κ). δ1, λ3, δ3, . . .
are the next ω Suslin cardinals after κ. S(λ2n+1) = Σ2n+1,
S(δ2n+1) = Σ2n+2.
20
ω
• cof(κ) > ω and Γ0 (Steel pointclass) not closed under ∃ω (type
II, III).
ω
Let Σ0 = ∃ω Γ0. Then scale(Γ0), scale(Σ0) with norms
into κ and scale(Π2n+1), scale(Σ2n+2) with norms into δ2n+1 =
δ(Π2n+1). λ1, δ1, λ3, . . . are the next ω Suslin cardinals after
κ. δ2n+1 = (λ2n+1)+, where cof(λ2n+1) = ω and S(λ2n+1) =
Σ2n+1, S(δ2n+1) = Σ2n+2.
ω
• cof(κ) > ω and Γ0 is closed under ∃ω (type IV).
Then scale(Γ0) with norms into
S κ. Let Λ = Λ(Γ0, κ) be the
Martin pointclass. Let Σ0 = ω Λ. Let λ1 = o(Λ). Then
scale(Π2n+1), scale(Σ2n+2) with norms into δ2n+1 = δ(Π2n+1).
λ1, δ3, λ3 are the next ω Suslin cardinals after κ. δ2n+1 =
(λ2n+1)+, where cof(λ2n+1) = ω and S(λ2n+1) = Σ2n+1, S(δ2n+1) =
Σ2n+2.
Steel gives a more detailed analysis assuming V = L(R).
21
Consider the first ω1 Suslin cardinals.
Let δ 1α = δ(Σ1α) (α ≥ 1).
At limit stages we are in type I, so we have the following picture.
···
δ 11
···
κω
δ 1ω+1 · · ·
λ3
δ 13
···
λω+3 δ 1ω+3
=
λ1
λω+1
The First ω1 Suslin Cardinals
22
• Compute the δ 1α.
• Use this analysis to determine the cardinal structure.
General Plan:
Establish the strong partition relation on the δ 1α, for α odd, and
use this to describe the cardinal structure.
The combinatorics involved is given by the notion of a description.
To get the strong partition relation on the odd δ 1α, we analyze the
measures on δ 1α, assuming such an analysis below δ 1α. This, by an
argument of Kunen, gives an analysis of the subsets of δ 1α which gives
a coding of the subsets of δ 1α sufficient to give the strong partition
relation on δ 1α (using a general theorem of Martin).
The measure analysis below δ 1α is enough to get the weak partition
relation on δ 1α which is enough to do the description analysis at
23
δ 1α which (1) computes the cardinal structure below λα+2 and (2)
analyzes the measures at δ 1α.
The arguments for (1) and (2) are very similar and use the same
combinatorics (i.e., same descriptions).
For simplicity, we concentrate here on (1), and show how to generate the cardinal structure assuming the strong partition relations
on the odd δ 1α. The argument for the measure analysis are similar.
Note that no new measures arise at limit stages since α < ω1.
So, assume the δ 1α, α odd, have the strong partition property.
24
Partition Properties and Types
Recall the Erdös-Rado partition notation.
κ → (κ)λ: For every partition P : κλ → {0, 1} there is a homogeneous set H ⊆ κ of size κ.
We say κ has the weak partition property if κ → (κ)λ for all λ < κ
and κ has the strong partition property if κ → (κ)κ.
More useful is the c.u.b. reformulation of the partition properties.
Definition. A function f : λ → On is of uniform cofinality ω if
there is a function g : λ × ω → On which is increasing in the second
argument and such that for all α < λ:
f (α) = sup g(α, n)
n
25
Definition. We say f is of the correct type if f is increasing, everywhere discontinuous (f (α) > supβ<α f (β) for limit α), and of
uniform cofinality ω.
c.u.b.
We say κ −→ (κ)λ if for every partition P of the function f : λ →
κ of the correct type, there is a c.u.b. C ⊆ κ homogeneous for P.
Fact.
c.u.b.
κ −→ (κ)λ ⇒ κ → (κ)λ
c.u.b.
κ → (κ)ω·λ ⇒ κ −→ (κ)λ
Proof. For the first fact, let P be a partition of the functions g : λ →
κ. In particular, P partitions functions of the correct type; let C
be c.u.b. and homogeneous for this subpartition. Define h : κ → κ
by h(α) = ω th element of C greater than supβ<α h(β). Let H =
ran(h), then H is homogeneous for P.
26
For the second fact, let P be a partition of the functions f : λ → κ
of the correct type. Let P 0 be the partition of increasing g : ω·λ → κ
given by P 0(g) = P(f ) where f (α) = supn g(α, n). Let H be
homogeneous for P 0 and let C be the set of limit points of H. C is
homogeneous for P.
More generally:
Definition. Let g : λ → On. We say f : λ → On is of uniform
cofinality g if there is a function
f 0 : {(α, β) : α < λ ∧ β < g(α)} → On
which is increasing in the second argument and such that f (α) =
supβ f 0(α, β).
If g is the constant ρ function, then we say f has uniform cofinality
ρ.
27
If µ is a measure on λ, we have the notion of f being of uniform
cofinality g almost everywhere w.r.t. µ.
If κ has strong partition property, we have partition property for
functions f : κ → κ of type g (increasing, discontinuous, of uniform
cofinality g). Given a measure on κ, this defines a measure on jµ(κ).
Analysis at δ 11 = ω1, Trivial Descriptions.
We start with weak partition property on ω1. [Use simple coding
of functions f : α → ω1 for α < ω1. Let π : ω → α be a bijection.
x ∈ ω ω codes partial function fx given by fx(π(n)) = |xn| if xn ∈
WO, undefined otherwise.]
From this we get that sets of the form C n form a base for a measure
on (ω1)n. We denote this W1n. It is the n-fold product of the normal
measure W11 on ω1.
28
Definition. A trivial description is an integer d ∈ ω. The set D
of trivial description is ordered in the usual ordering on ω. The
lowering operator L is defined on all d ∈ D except the minimal
description d = 0. L(d) = d − 1.
Let Dn be those trivial descriptions d < n.
Interpretation: If f : n → ω1, and d ∈ Dn, define h(f ; d) = f (d).
Let (W1n; d) = [f 7→ (f ; d)]W1n .
If g : ω1 → ω1, let also:
(g; f ; d) = g(f (d)), (g; W1n; d) = [f 7→ (g; f ; d)]W1n .
Kunen Tree Aside from partition property of ω1, we need the
Kunen Tree.
29
For x ∈ ω ω , let <x= {(n, m) : hn, mi) = 1}.
LO = {x : <x is a linear order},
WO = {x : <x is a wellorder}.
Let S be the Shoenfield tree on ω × ω1 with WO = p[S].
(s, α
~ ) ∈ S iff s doesn’t violate being in LO and ∀i, j ≤ |s| (s(hi, ji) =
1 → αi < αj ).
With a small patch-up. can assume S is homogeneous, i.e., s
determines the order-type of α
~ . Also, α0 > α1, . . . , αn−1.
Theorem (Kunen). There is a tree T on ω × ω1 with the following property. For any f : ω1 → ω1 there is an x ∈ ω ω such that
Tx is wellfounded and for all infinite ordinals α < ω1 we have
f (α) < |Tx α|.
30
Proof. If f : ω1 → ω1, play the game where I plays out x, II plays
out y, and II wins the run iff
x ∈ WO → (y ∈ WO ∧ |y| ≤ f (|x|)).
II has a winning strategy by boundedness. This suggests the following definition.
V is the tree on ω × ω × ω1 × ω × ω given by:
(s,t, α
~ , u, v) ∈ V ↔ ∃σ, x, y, z extending s, t, u, v
[σ(x) = y ∧ (t, α
~) ∈ S ∧
∀i y(hz(i), z(i + 1)i) = 1]
For f and σ as above, Vσ is wellfounded and for any infinite α,
|Vσ α| ≥ f (α).
Can identify V with a tree on ω × ω1.
An easy partition argument shows:
If f : (ω1)n → ω1 is such that f (~
α) < αi for almost all α
~ , then
~ f (~
α) < NC (αi−1).
there is a c.u.b. C ⊆ ω1 such that ∀∗α
31
NC (β) = least element of C greater than β.
Translating we have:
Main Theorem: If α < (W1n; d), then there is a g = NC such
that α < (g; W1n; L(d)).
Fix x ∈ ω ω such that g(β) ≤ |Tx β| for almost all β.
Then
(g; W1n; L(d)) ≤ (β 7→ |Tx β|; W1n; L(d))
< (W1n; L(d))+
So, (W1n; d) ≤ (W1n; L(d))+.
Corollary. jW1n (ω1) ≤ ωn+1.
32
To get the lower bound we use:
Theorem (Martin). Assume κ → (κ)κ. Then for any measure
µ on κ, jµ(κ) is a cardinal.
Also, if m < n then jW1m (ω1) < jW1n (ω1). In fact, jW1m (ω1) embeds
into (W1m+1; d), where d = m + 1.
So we have jW1n (ω1) = ωn+1.
Recall Shoenfield tree for a Σ12 set is weakly homogeneous with
measures of the form W1n. Homogeneous tree construction gives
that every Π12, and hence also every Σ13 set is λ-Suslin where λ =
supn jW1n (ω1) = ωω .
So, λ3 ≤ ωω and δ 13 ≤ ωω+1.
Since cof(λ3) = ω we must have λ3 = λ = ωω and δ 13 = ωω+1.
33
Types of Functions on ω1
Fix n, and a permutation π of {1, 2, , . . . , n} beginning with n.
π = (n, i2, . . . , in).
We say f : (ω1)n → ω1 is ordered by π if on a measure one set we
have:
f (α1, . . . , αn) < f (β1, . . . , βn) ↔
(αn, αi2 , . . . , αin ) <lex (βn, βi2 , . . . , βin ).
Fact. If f depends on all its arguments, then for some π, f is
ordered by π.
If f : (ω1)n → ω1 then the type of f is determined by π and the
possible uniform cofinalities:
34
• There is a measure one set on which f is continuous (i.e., f C n
is continuous).
• f has uniform cofinality ω (of the correct type).
• f (α1, . . . , αn) has uniform cofinality αi.
Example. Show that the uniform cofinalities g(~
α) = αi and g(~
α) =
αj are distinct for i 6= j.
Proof. Suppose f1 : {(~
α, β) : β < αi} → ω1 induces f , and likewise
for
f2 : {(~
α, β) : β < αj }.
Consider the partition P: partition
α1 < · · · < αi−1 < β1 < αi < · · ·
< αj−1 < β2 < αj < · · · < αn
according to whether f1(~
α, β1) < f2(~
α, β2). Neither side can be
homogeneous, a contradiction.
35
We introduce a canonical family of measures for each regular cardinal below ℵω1 .
Let πn be the permutation
πn = (n, 1, 2, . . . , n − 1).
Definition. S1n is the measure on ωn+1 induced by functions f :
(ω1)n → ω1 ordered by πn and of the correct type (and the measure
W1n).
S11 is the ω-cofinal normal measure on ω2.
Fact. The family of measures S1n dominate all the measures on ωω .
More precisely: For any measure µ on ωω there is an n and a c.u.b.
C ⊆ δ 13 such that for all α ∈ C with cof(α) = ω2,
jµ(α) ≤ jS1n (α).
36
Example. We show this for µ = ν × ν, where ν is the measure
corresponding to π = (3, 2, 1).
Proof. We take n = 3. We define an auxiliary measure M = W11 ×
W11 × S12.
Given f : <3→ ω1 of the correct type and given η1 < η2 <
ω1 and h3 representing (η1, η2, [h3]) ∈ ω1 × ω1 × ω3, we define
g1, g2 : (ω1)3 → ω1 by:
gi(α1, α2, α3) = f (ηi, h3(α1, α2), α3).
Let α ∈ C, the set of ordinals < δ 13 closed under ultrapowers by
measures on ωω .
Define π : jµ(α) → jS 3 (α) as follows.
1
π([F ]µ) = [G]S 3 .
1
37
G([f ]W 3 ) =
1
[(η1, η2, [h3]) 7→ G(f, η1, η2, h3)]M.
G(f, η1, η, h3) = F ([g1], [g2]), where gi = gi(f, η1, η2, h3) as above.
The following observations complete the proof.
(1) For any fixed f of the correct type, fixed η1, η2, h3 with η1 <
η2 < h3(γ, β) for all γ < β and all α1 < α2 < α3 in a c.u.b. set
closed under h3(0) we have that gi is welldefined and of order
π.
~ iff α3 ≤ β3 (the type of the product
Also, g1(~
α) < g2(β)
measure µ.
(2) For any fixed f , [gi] only depends on η1, η2, [h3].
(3) If [f1] = [f2], then M almost all η1, η2, [h3] we have that
[gi1] = [gi2], where gi1 uses f1 and likewise for gi2.
38
(4) If µ(A) = 1, then
∀∗f ∀∗η1, η2, h3 ([g1], [g2]) ∈ A.
(5) ∀∗f ∃(G1, G2) ∀∗η1, η2, h3 (g1 ≤ G1 ∧ g2 ≤ G2).
(1)-(4) give that π is welldefined, (5) and α ∈ C give that π([F ]) <
jS 3 (α).
1
39
Notation Convention
Suppose θ ∈ On, µ1, . . . , µn are measures, and P ⊆ On. We write
∀∗µ1 α1 · · · ∀∗µn αn P (θ(α1, . . . , αn))
to abbreviate the following:
If we fix α1 7→ θ(α1) representing θ in the ultrapower by µ1, then
for µ1 almost all α1 it is the case that:
If we fix α2 7→ θ(α1, α2) representing θ(α1) with respect to µ2,
then for µ2 almost all α2 it is the case that:
..
If we fix αn 7→ θ(α1, . . . , αn) representing θ(α1, . . . , αn−1) with
respect to µn, then for µn almost all αn we have P (θ(α1, . . . , αn)).
40
Example. Use trivial descriptions to compute jS1m (ωn) = ωk where
n−1
n−1
].
+ ··· +
k = 1 + 2[
m
1
We describe the cardinals below jS1m (ωn). We define a set S of
special instances of non-trivial descriptions.
Definition. For m, n ≥ 1, let Sm,n be the set of tuples of the form
d = (dm, d1, d2, . . . , d`) or d = (dm, d1, d2, . . . , d`)s where the di are
trivial descriptions in Dn−1 (i.e., 1 ≤ di ≤ n − 1), ` < m, and
d1 < d2 < · · · < d` < dm.
We write d = (dm, d1, d2, . . . , d`)(s) to denote that s may or may
not appear.
Given d ∈ Sm,n, we associate an ordinal (d; S1m) to d defined as
follow.
41
We represent (d; S1m) with respect to the measure S1m by the function
[f ] 7→ (d; f ),
for f : (ω1)m → ω1 of the correct type.
(d; f ) < ωn is the ordinal represented with respect to W1n−1 by
the function
(α1, . . . , αn−1) 7→ (d; f )(~
α).
Finally,
(d; f )(~
α) = f (` + 1)(αd1 , αd2 , . . . , αd` , αdm ).
if s does not appear in d.
If s appears in d, let
42
(d; f )(~
α) = f s(` + 1)(αd1 , αd2 , . . . , αd` , αdm )
= sup f (` + 1)(αd1 , αd2 , . . . , β, αdm ).
β<αd
`
There are 2[
n−1
1
+ ··· +
n−1
m
] many such ordinals (d; S1m).
Claim. These are precisely the cardinals below jS1m (ωn).
The descriptions in Sm,n are ordered as follows.
Set (dm, d1, . . . , d`)(s) < (d0m, d01, . . . , d0`0 )(s) iff one of the following
holds:
(1) There is a least place of disagreement in the description sequences, say di 6= d0i, and di < d0i.
~
(2) d~ is an initial segment of d~0, and s appears in d.
43
~ and s does not appear in d~0.
(3) d~0 is an initial segment of d,
Let L(dm, d1, . . . , d`)(s) be the largest sequence which is less than
(dm, d1, . . . , d`)(s).
We describe the L operation explicitly:
• If ` = m − 1 then L(dm, d1, . . . , d`) = (dm, d1, . . . , d`)s.
• If ` = m − 1 then L(dm, d1, . . . , d`)s = (dm, d1, . . . , L(d`)) if
L(d`) = d` − 1 is defined and > d`−1 (if ` > 1). Otherwise,
L(dm, d1, . . . , d`)s = (dm, d1, . . . , dl−1)s.
• If ` < m − 1, then L(dm, d1, . . . , d`) = (dm, d1, . . . , d`, d`+1)
where d`+1 = L(dm) = dm − 1 provided dm − 1 > d` (if ` ≥ 1).
Otherwise set L(dm, d1, . . . , d`) = (dm, d1, . . . , d`)s.
44
Example. m = 3, n = 5 (i.e., jS 3 (ω5))
1
d = (4)
L3(d) = (4, 2)
L6(d) = (4, 2)s
L9(d) = (4, 1, 3)s
L12(d) = (4, 1)s
L15(d) = (3, 2)
L18(d) = (3, 1, 2)
L21(d) = (3)s
L24(d) = (2, 1)s
L27(d) = (1)s
L(d) = (4, 3)
L4(d) = (4, 2, 3)
L7(d) = (4, 1)
L10(d) = (4, 1, 2)
L13(d) = (4)s
L16(d) = (3, 2)s
L19(d) = (3, 1, 2)s
L22(d) = (2)
L25(d) = (2)s
L2(d) = (4, 3)s
L5(d) = (4, 2, 3)s
L8(d) = (4, 1, 3)
L11(d) = (4, 1, 2)s
L14(d) = (3)
L17(d) = (3, 1)
L20(d) = (3, 1)s
L23(d) = (2, 1)
L26(d) = (1)
So, jS 3 (ω5) = ω29.
1
The upper bound follows from the following lemma.
45
Lemma. For d ∈ Sm,n non-minimal, (d; S1m) ≤ (L(d); S1m)+. If
d is L-minimal, then (d; S1m) ≤ ω1.
Proof. We consider the case d = (4, 1) in the above example. L(d) =
(4, 1, 3).
Let θ < (d; S1m). Then,
∀∗S1m [f ] θ([f ]) < (d; f ).
Thus,
∀∗S1m [f ] ∀∗W n−1 α
~ θ(f )(~
α) < (d; f )(~
α) = f (2)(α1, α4)
1
= sup f (α2, β, α4).
β<α4
So,
46
~ ∃β < α4 θ(f )(~
α) < f (α1, β, α4).
∀∗S1m [f ] ∀∗W n−1 α
1
A partition argument shows that there is a g : ω1 → ω1 such that
~ θ(f )(~
α) < f (α1, g(α3), α4).
∀∗S1m [f ] ∀∗W n−1 α
1
Fix x ∈ ω ω such that |Tx α| > g(α) almost everywhere.
Thus,
~ θ(f )(~
α) < f (α1, |Tx α3|, α4).
∀∗S1m [f ] ∀∗W n−1 α
1
It follows that there a θ0 < (L(d); S1m) such that
∀∗S1m [f ] ∀∗W n−1 α
~ θ(f )(~
α) = f (α1, |Tx α3(θ0(f )(~
α))|, α4).
1
where |Tx α(β)| denotes the rank of β in Tx α.
47
The map δ 0 7→ δ defined by
~
∀∗S1m [f ] ∀∗W n−1 α
1
δ(f )(~
α) = f (α1, |Tx α3(δ 0(f )(~
α))|, α4).
defines a map from (L(d); S1m) onto θ, so θ < (L(d); S1m)+.
The lower bound follows from the following lemma.
Lemma. (d; S1m) ≥ ω|d|+1, where |d| denotes the rank of d in the
L ordering.
Proof. Let k be the rank of d in the L-order. Thus there are k
elements of Sm,n below d, say d1 < · · · < dk .
We define an embedding π from ωk+1 = jW k (ω1) into (d, S1m).
1
48
For g : (ω1)k → ω1, define π([g]W k ) = θ where
1
~
∀∗S1m f ∀∗W n−1 α
1
θ(f )(~
α) = g((d1; f )(~
α), . . . , (dk ; f )(~
α)).
That this works follows from the following observations.
(1) For fixed g, θ(f ) depends only on [f ]W1m , since if [f1] = [f2] then
almost all α
~ will be in a c.u.b. C such that f1 C n−1 = f2 C n−1.
(2) If [g1]W k = [g2]W k , and say g1 C k = g2 C k , then S1m almost
1
1
all [f ] are represented by f : <m→ C of the correct type. Thus,
∀∗S m f ∀∗W n−1 α
~ we have
1
1
g1((d1; f )(~
α), . . . , (dk ; f )(~
α)) =
g2((d1; f )(~
α), . . . , (dk ; f )(~
α)).
49
Thus, θ depends only on [g]W k .
1
(3) For any fixed g, almost all f have range in a c.u.b. set closed
under g(0). Thus,
θ(f )(~
α) = g((d1; f )(~
α), . . . , (dk ; f )(~
α)) < (d; f )(~
α).
So, π([g]) < (d; S1m).
(1) and (2) show π is well-defined, and (3) show π embeds jW k
1
m
into (d; S1 ).
50
Non-trivial Descriptions
To motivate non-trivial descriptions, we consider the following
problem.
Problem.
Compute jS m1 ◦ jS m2 ◦ · · · ◦ jS mt (ωn).
1
1
1
On the one hand, our previous formula computes this value.
Example. jS 2 ◦ jS 2 (ω3) = jS 2 (ω7) = ω43.
1
1
1
We wish, however, to analyze the iterated ultrapower directly. This
leads to the general (next level) notion of description.
Set-Up: Given a finite sequence of measures S1m1 , . . . , S1mt and an
integer n − 1 (corresponding to jW n−1 (ω1) = ωn).
1
51
We will define for each such sequence of measures and n − 1 a set
of descriptions
Dn−1(S1m1 , . . . , S1mt ).
Slightly more generally, allow a finite sequence of measures K1, . . . , Kt
each of the form S1m or W1m. So, we define
Dn−1(K1, . . . , Kt).
Such a d will give an ordinal (d; K1, . . . , Kt) as follows.
(d; K1, . . . , Kt) is represented w.r.t. K1 by the function [h1] 7→
(d; h1; K2, . . . , Kt).
Here h1 : <m1 → ω1 of the correct type if K1 = S1m1 , and h1 : m1 →
ω1 if K1 = W1m1 .
52
(d; h1; K2, . . . , Kt) is represented w.r.t. K2 by the function [h2] 7→
(d; h1, h2, K3, . . . , Kt).
Finally, (d; h1, . . . , ht) < ωn is represented with respect W1n−1 by
the function
α ) < ω1 .
(α1, . . . , αn−1) 7→ (d; h1, . . . , ht)(~
It remains to define D and the interpretation (d; h1, . . . , ht)(~
α).
Main Point: Allow composition of the h’s.
Definition of D = Dn−1(K1, . . . , Kt).
D is defined through the following cases. We also define a value
k(d) ∈ {1, . . . , t} ∪ {∞}.
• (1) We allow d = ·i for 1 ≤ i ≤ n − 1. Set k(d) = ∞.
53
α) = αi.
Interpretation: (d; ~h)(~
m
• (2) If Kk = W1 k , we allow d = (k; i) for 1 ≤ i ≤ mk . Set
k(d) = k.
Interpretation: (d; ~h)(~
α) = hk (i).
(1) and (2) are called basic descriptions.
• (3) If Kk = S1m we allow
d = (k; dm, d1, . . . , d`)(s)
where k(d1), . . . , k(d`), k(dm) > k and d1 < · · · < d` < dm (defined
below).
Interpretation:
(s)
(d; ~h)(~
α) = hk (` + 1)((d1; ~h)(~
α), . . . , (d`; ~h)(~
α), (dm; ~h)(~
α)).
54
The descriptions from (3) are called non-basic descriptions.
Definition of Ordering:
d < d0 iff
∀∗h1, . . . , ht ∀∗α
~ (d; ~h)(~
α) < (d0; ~h)(~
α).
Can describe the < relation directly. It is also generated by a
lowering operator L as before. We describe L directly.
We define for k ∈ {1, . . . , t} ∪ {∞} and d ∈ D with k(d) ≥ k a
partial lowering Lk (d). We will take L(d) = L1(d).
Definition of Lk We define Lk through the following cases.
(1) k = ∞. In this case, d = ·i. We define Lk (d) = i − 1 if i > 1
and otherwise d is Lk minimal.
55
In the remaining cases, 1 ≤ k ≤ t.
(2) Kk = W1m. If k(d) = k, then d = (k, i) for some 1 ≤ i ≤ m.
We set Lk (d) = i − 1 unless i = 1 in which case d is Lk -minimal. If
k(d) > k, then Lk (d) = Lk+1(d) unless d in Lk+1 minimal in which
case Lk (d) = (k, m).
(3) Kk = S1m and k(d) > k. If d is Lk+1-minimal then d is also
Lk -minimal. Otherwise set Lk (d) = (k; Lk+1(d)).
(4) Kk = S1m and k(d) = k. So, d = (k; dm, d1, . . . , d`)(s).
(a) ` = m−1 and s does not appear. Lk (d) = (k; dm, d1, . . . , d`)s.
(b) ` = m−1 and s does appears. Lk (d) = (k; dm, d1, . . . , Lk+1(d`))
if Lk+1(d`) is defined and > d`−1 (if ` > 1). Otherwise Lk (d) =
(k; dm, d1, . . . , d`−1)s.
56
If m = 1 (so ` = 0) or if ` = 1 and Lk+1(d`) is not defined, set
Lk (d) = dm.
(c) ` < m−1 and s does not appear. Lk (d) = (k; dm, d1, . . . , d`, Lk+1(dm))
if Lk+1(dm) is defined and > d` (if ` ≥ 1). Otherwise, Lk (d) =
(k; dm, d1, . . . , d`)s.
(d) ` < m − 1 and s appears. Lk (d) = (k; dm, d1, . . . , Lk+1(d`))s
if Lk+1(d`) is defined and > d`−1 (if ` > 1). Otherwise, Lk (d) =
(k; dm, d1, . . . , d`−1)s if ` > 1 and = dm for ` = 1.
57
Claim. The (d; S1m1 , . . . , Skmt ) correspond to the cardinals below
jS m1 ◦ jS m2 ◦ · · · ◦ jS mt (ωn).
(Where d ∈
1
1
1
Dn−1(S1m1 , . . . Stmt ))
Return to example jS 2 ◦ jS 2 (ω3).
1
1
58
d = (1; (2; ·2))
L(d) = (1; (2; ·2); (2; ·2, ·1))
L3(d) = (1; (2; ·2); (2; ·2, ·1)s)
L2(d) = (1; (2; ·2); (2; ·2, ·1))s
L4(d) = (1; (2; ·2); (2; ·2, ·1)s)s
L5(d) = (1; (2; ·2); ·2)
L7(d) = (1; (2; ·2); (2; ·1))
L6(d) = (1; (2; ·2); ·2)s
L8(d) = (1; (2; ·2); (2; ·1))s
L9(d) = (1; (2; ·2); ·1)
L11(d) = (2; ·2)
L10(d) = (1; (2; ·2); ·1)s
L12(d) = (1; (2; ·2, ·1))
L13(d) = (1; (2; ·2, ·1); (2, ·2, ·1)s)
L15(d) = (1; (2; ·2, ·1); ·2)
L14(d) = (1; (2; ·2, ·1); (2, ·2, ·1)s)s
L16(d) = (1; (2; ·2, ·1); ·2)s
L17(d) = (1; (2; ·2, ·1); (2; ·1))
L19(d) = (1; (2; ·2, ·1); ·1)
L18(d) = (1; (2; ·2, ·1); (2; ·1))s
L20(d) = (1; (2; ·2, ·1); ·1)s
L21(d) = (2; ·2, ·1)
L23(d) = (1; (2; ·2, ·1)s, ·2)
L22(d) = (1; (2; ·2, ·1)s)
L24(d) = (1; (2; ·2, ·1)s, ·2)s
L25(d) = (1; (2; ·2, ·1)s, (2; ·1))
L26(d) = (1; (2; ·2, ·1)s, (2; ·1))s
L27(d) = (1; (2; ·2, ·1)s, ·1)
L28(d) = (1; (2; ·2, ·1)s, ·1)s
L29(d) = (2; ·2, ·1)s
L30(d) = (1; ·2)
L31(d) = (1; ·2, (2; ·1))
L32(d) = (1; ·2, (2; ·1))s
L33(d) = (1; ·2, ·1)
L34(d) = (1; ·2, ·1)s
L35(d) = ·2
L36(d) = (1; (2; ·1))
L37(d) = (1; (2; ·1), ·1)
L38(d) = (1; (2; ·1), ·1)s
L39(d) = (2; ·1)
L41(d) = ·1
L40(d) = (1; ·1)
59
So, we again have jS 2 ◦ jS 2 (ω3) = ω43.
1
1
Remark. The iterated ultrapower is not the same as the ultrapower
by the product measure in the AD context. For example, jS 2 ◦
1
jS 2 (ω3) = ω43 but jS 2×S 2 (ω3) = ω11.
1
1
1
The proof of the lower bound is exactly as before (using now the
non-trivial descriptions).
The upperbound follows from the following lemma.
For g : ω1 → ω1 define θ = (g; d; K1, . . . , Kt) by:
∀∗h1, . . . , ht ∀∗α
~
θ([f1], . . . , [ft])(~
α) = g((d; f1, . . . , ft)(~
α)).
60
Lemma. If θ < (d; K1, . . . , Kt), then there is a g : ω1 → ω1 such
that θ < (g; L(d); K1, . . . , Kt).
~ ≤ (L(d); K)
~ +.
It follows that (d; K)
Example of proof. Consider
d = L16(−) = (1; (2; ·2, ·1), ·2)s
L(d) = L17(−) = (1; (2; ·2, ·1), (2; ·1)).
from the above example.
Suppose θ < (d; S12, S12). Then,
∀∗S 2 [h1] ∀∗S 2 [h2] ∀∗W 2 α1, α2
1
1
1
θ([h1], [h2])(~
α) < (d; h1, h2)(~
α)
= sup h1(β, h2(α1, α2))
β<α2
So,
61
∀∗S 2 [h1] ∀∗S 2 [h2] ∀∗W 2 α1, α2∃β < α2
1
1
1
θ([h1], [h2])(~
α) < h1(β, h2(α1, α2))
Hence,
∀∗S 2 [h1] ∀∗S 2 [h2] ∃g : ω1 → ω1 ∀∗W 2 α1, α2
1
1
1
θ([h1], [h2])(~
α) < h1(g(α1), h2(α1, α2))
For any h1 and ordinal θ(h1) such that ∀∗S 2 [h2] ∃g · · · , consider
1
the partition
P(h1): We partition pairs (h2, g) where h2 : <2→ ω1 is of the
correct type, g is of the correct type, and
h2(0)(γ) < g(γ) < h2(0, γ + 1)
for all γ according to whether
62
∀∗S 2 [h2] ∀∗W 2 α1, α2
1
1
θ(h1)(h2)(~
α) < h1(g(α1), h2(α1, α2))
On the homogeneous side this must hold. This gives:
∀∗S 2 h1 ∃c.u.b.C ⊆ ω1 ∀∗S 2 [h2] ∀∗W 2 α1, α2
1
1
1
θ(h1)(h2)(~
α) < h1(NC (h2(0)(α1)), h2(α1, α2))
where NC (β) =least element of C greater than β.
Next consider the partition
P: We partition h1, g : <2→ ω1 of the correct type with h1(α1, α2) <
g(α1, α2) < Nh1 (α1, α2) according to whether
63
∀∗S 2 [h2] ∀∗W 2 α1, α2
1
1
θ(h1)(h2)(~
α) < g(h2(0)(α1), h2(α1, α2))
On the homogeneous side the stated property holds. Fixing C ⊆ ω1
homogeneous we get:
∀∗S 2 [h2] ∀∗W 2 α1, α2
1
1
θ(h1)(h2)(~
α) < NC (h2(0)(α1), h2(α1, α2))
= (NC ; L(d); h1, h2)(~
α)
and we are done.
64
Analysis between δ 13 and δ 15
We used trivial descriptions to analyze the cardinals below δ 13.
Now we use the (non-trivial) descriptions to analyze the cardinal
structure below δ 15.
Recall that for a sequence of measures K1, . . . , Kt, with each Kk =
m
m
~ of descriptions
W1 k or S1 k , and each n, we have a set Dn(K)
defined.
Assume inductively the weak partition relation on δ 13 (i.e., analysis
of measures below δ 13).
Definition. W3m is the measure on δ 13 induced by the weak partition relation of δ 13, functions f : ωm+1 → δ 13 of the correct type, and
the measure S1m on ωm+1.
65
The measure W3m on δ 13 dominate all of the measures on δ 13 in the
following sense.
Fact. For any measure µ on δ 13, there is an m such that jµ(δ 13) <
jW3m (δ 13).
The homogeneous tree construction show that λ5 = supµ(δ 13), and
so λ5 = supm jW3m (δ 13).
We define an ordinal (d; W3n; K1, . . . , Kt) < jW3n (δ 13) for d ∈
~
Dn(K).
We define this in the usual iterated way, where for f : ωn+1 → δ 13,
h1, . . . , ht (hi : <i→ ω1) we have:
(d; f ; h1, . . . , ht) = f ((d; h1, . . . , ht)).
66
We also consider objects of the form ds and define:
(ds; f ; h1, . . . , ht) = sup{f (β) : β < f ((d; h1, . . . , ht))}.
These are well-defined provided d(s) satisfies the following.
Definition. d ∈ Dn(K1, . . . , Kt) satisfies condition C if ∀∗~h (d; ~h)
is almost everywhere of the correct type. ds satisfies C if ∀∗~h (d; ~h)
is the supremum of ordinals represented by functions of the correct
type.
We set L0((d)) = (d)s, and L0((d)s) = (L(d)). Let L((d)(s)) to the
least iterate of L0 satisfying C.
Theorem. The cardinals below λ5 are precisely those of the
~ with d(s) satisform (d(s); W3n, K1, . . . , Kt) for some d ∈ Dn(K)
fying condition C.
67
~ is given by the rank of
The cardinal corresponding to (d; W3m; K)
this tuple in the ordering generated by the relation:
((d)(s); W3m,K1, . . . , Kt) <
(L((d)(s)); W3m; K1, . . . , Kt, Kt+1).
The upper bound follows from the following theorem.
For g : δ 13 → δ 13, define (g; d; W3n, K1, . . . , Kt) by:
∀∗W3n [f ] ∀∗h1, . . . ht
(g; d; f, h1, . . . , ht) = g(f (s)((d; h1, . . . , ht))).
~ then there is a g : δ 13 → δ 13 such
Theorem. If θ < (d; W3n, K),
~
that θ < (g; d; W3n; K).
68
The lower bound follows by embedding arguments roughly similar
to those of the iterated ultrapower computation.
Example. Consider ((d); W32; S12, S12) where d = L26(1; (2; ·2)) =
L26(dM ) = (1; (2; ·2, ·1)s, (2; ·1))s (considered previously).
69
We have:
|((d); W32, S12, S12)| = sup |(L26(dM ))s; W32, S12, S12, K3)| + 1
=
=
=
=
=
=
K3
~
sup |((L27(dM )); W32, S12, S12, K)|
~
K
~
sup |((L29(dM )); W32, S12, S12, K)|
~
K
~
sup |((L30(dM )); W32, S12, S12, K)|
~
K
~
sup |((L31(dM )); W32, S12, S12, K)|
~
K
~
sup |((L33(dM )); W32, S12, S12, K)|
~
K
~
sup |((L36(dM )); W32, S12, S12, K)|
~
K
ω
+ω+1
+ω+ω+1
+ω+ω+ω+1
+ ωω + ω + ω + ω + 1
+ ω + ωω + ω + ω + ω + 1
+ ωω + ω + ωω + ω + ω + ω + 1
=ω ·2+ω·3+1
That is, ((d); W32, S12, S12) = ℵωω ·2+ω·3+1.
70
More on the Cardinal Structure
The three normal measure on δ 13 correspond to the three regular
cardinals below δ 13, namely ω, ω1, ω2.
The three regular cardinals between δ 13 and δ 15 correspond to jµ(δ 13)
for µ one of these normal measures.
A description computation as above computes these to be:
jµω (δ 13) = δ 14 = ℵω+2
jµω1 (δ 13) = ℵω·2+1
jµω2 (δ 13) = ℵωω +1
Also,
71
jW3n (δ 13) = ℵωωn +1.
So,
λ 5 = ℵω ω ω ,
δ 15 = ℵωωω +1
We can easily read off the cofinality from the description.
Example. Consider again ((d); W32; S12, S12) where d = L26(1; (2; ·2)) =
(1; (2; ·2, ·1)s, (2; ·1))s.
Ordinals of the form (g; (d)s; W32, S12, S12) for g : δ 13 → δ 13 are cofinal in ((d); W32, S12, S12).
Now, ∀∗W n f ∀∗h1, h2, we are evaluating g at f s((d; h1, h2)) =
3
supβ<(d;h1,h2) f (β). This has the same cofinality as (d; h1, h2).
Now (d; h1, h2) has uniform cofinality (α1, α2) 7→ α1, and so
(d; h1, h2) has cofinality ω1.
72
Thus, (g; (d)s; W3n, S12, S12) depends only on [g]µω1 .
This gives a cofinal embedding from jµω1 (δ 13) into ((d); W3n, S12, S12).
Thus, cof(ℵωω ·2+ω·3+1) = ℵω·2+1.
Using the “linear” theory (described below), we can efficiently compute the cofinalities of all the cardinals. For example, below δ 15 we
have:
Theorem. Suppose δ 13 < ℵα+1 < δ 15. Let α = ω β1 + · · · + ω βn ,
where ω ω > β1 ≥ · · · ≥ βn be the normal form for α. Then:
• If βn = 0, then cof(κ) = δ 14 = ℵω+2.
• If βn > 0, and is a successor ordinal, then cof(κ) = ℵω·2+1.
• If βn > 0 and is a limit ordinal, then cof(κ) = ℵωω +1.
73
In general δ 12n+1 = ℵω(2n−1)+1 where ω(0) = 1 and ω(n + 1) =
ω ω(n).
There are 2n+1 −1 many regular cardinals below δ 12n+1. The regular
cardinals between δ 12n−1 and δ 12n+1 correspond to the ultrapowers of
δ 12n+1 by the normal measures on δ 12n+1, which correspond to the
regular cardinals below δ 12n+1.
There is a canonical family of measures associated to each regular
cardinal.
Families are:
W1m, S11,m, W3m, S31,m, S32,m, S33,m,
W5m, S51,m, . . . , S57,m, W7m, . . .
74
m
W2n+1
defined using weak partition relation on δ 12n+1, function
`,m
`,m
f : = dom(S2n−1
) of the correct type and the measure S2n−1
. Here
` = 2n − 1.
1,m
S2n+1
defines as S1m was defined using δ 12n+1.
`,m
S2n+1
for ` > 1 defined using the strong partition relation on δ 12n+1
functions F : δ 12n+1 → δ 12n+1 of the correct type and the measure µ
on δ 12n+1.
µ is the measure induced by the weak partition relation on δ 12n+1,
functions f : dom(ν) → δ 12n+1 and the measure ν, where ν is the
mth measure in the ` − 1st family.
75
General Descriptions:
The general level descriptions are defined inductively, and have
indices associated to them.
Trivial descriptions are level-1 descriptions. They have empty index. These analyze the measure on δ 11 and compute δ 13.
Level-2 descriptions are those of the form d = (k; dm, d1, . . . , d`)(s)
~ we associate the index W1n
we we defined previously. If d ∈ Dn(K),
n
to d i.e., (d = dW1 ). These analyze the measure on λ3.
The level-3 descriptions are those of the form (d) or (d)s for d a
level-2 description. These analyze the measures on δ 13 and compute
δ 15. They also have index W1n.
note: There is not much difference between the level-2 and level-3
desciptions; we could group them together.
76
Level-4 and 5 descriptions analyze the measures on λ5 and δ 15
~
respectively. They are defined relative to a sequence of measures K
~ for level-5.
where Ki ∈ W1m, . . . , S33,m for level-4, and (W5m, K)
They have indices of the form (W3m, S11,m1 , . . . , S11,mt ) (can use
W1mi ).
A basic description at this level is a level-3 description in
D(W3m, S11,m1 , . . . , S11,mt ).
77
Linear Analysis
We describe the cardinal structure without using descriptions or
iterated ultrapowers (with Khafizov, Löwe).
Need the description analysis to prove it works, however.
Need this to show that all descriptions actually represent cardinals.
Definition. Let Aα be the free algebra on α generators {Vβ }β<α
using the operations ⊕, ⊗.
A=
S
α Aα .
We assign an ordinal height o(v) to every term in A inductively as
follows.
78
Definition.
o(v0) = 0
o(vα) = ht(Aα) = sup{o(t) + 1 : t ∈ Aα}
o(s ⊕ t) = o(s) + o(t)
o(s ⊗ t) = o(s) · o(t)
ω
ω2
So, o(v0) = 0, o(v1) = 1, o(v2) = ω, o(v3) = ω , o(v4) = ω ,
n−2
ω
o(vn) = ω ω , o(vω ) = ω ω .
α
For α ≥ ω, o(vα) = ω ω .
We assign to each generator vα an order type ot(vα) and a measure
µ(vα) on this order-type.
This will generate an assignment of an order-type and a collection of measures (“germ”) to each general term as in the following
example.
79
Example. Consider the term
t = (((v3 ⊕ v2 ⊕ v1) ⊗ v4) ⊗ v2) ⊕ ((v3 ⊕ v2 ⊕ v1) ⊗ v1)
We can represent this as a tree as follows:
•
33
33
33
v2
v
1
222
22
2
v422 v3 v2 v1
22
22
v3 v2 v1
Suppose we know
o(v1) = 1,
o(v2) = ω1
o(v3) = ω2
o(v4) = ω3
µ(v1) = principal measure
µ(v2) = W11
µ(v2) = S11
µ(v2) = S12
80
Then ot(t) = (ω2 +ω1 +1)·ω3 ·ω1 +(ω2 +ω1 +1)·1 = ω3 ·ω1 +ω2 +ω1.
We identify ot(t) the ordering on tuples hi0, α0, . . . , ik , αk i where
(i0, . . . , ik ) corresponds to a terminal node in the tree, and for
~
(i0, . . . , i`) a node in the tree, α` < ot(v)), where v = v i is the
variable corresponding to this node.
~
µ(t) is the collection of measures µ(v i).
We set ot(vω(2n−1)) = δ 12n+1, µ(vω(2n−1)) = ω-cofinal normal measure on δ 12n+1.
For example, ot(vω ) = δ 13, ot(vωωω ) = δ 15.
To v2+ω(2n−1)+α (where α < ω(2n + 1)) we associate the measure
defined by the strong partition relation on δ 12n+1 and the measure ν
on δ 12n+1.
81
ν is the measure induced by the weak partition relation on δ 12n+1,
functions f : ot(t) → δ 12n+1 of the correct type, and the measure
(germ) µ(t).
(note: we order by reverse lexicographic order on the indices,
though it turns out this doesn’t matter).
Example. µ(vω+1) = ω-cofinal normal measure on δ 14, µ(vω+2) is
measure on ℵω+3. µ(vω·2) is the ω-cofinal normal measure on ℵω·2+1.
This assignment describes the cardinal structure as follows.
Theorem. Let t ∈ A with ot(t) < δ 12n+1. Then jν (δ 12n+1) =
ℵω(2n−1)+o(t)+1 where ν = ν(t) is the measure defined above.
Example. Consider the term
t = (((v3 ⊕ v2 ⊕ v1) ⊗ v4) ⊗ v2) ⊕ ((v3 ⊕ v2 ⊕ v1) ⊗ v1)
82
above.
Then jν(t)(δ 13) = ℵωω2 ·ω+ωω +ω+2.
83
A Collapsing Result
Theorem. Assume the non-stationary ideal on ω1 is ω2-saturated
and there are ω + 1 Woodin cardinals in V . Then there is a
κ < (ℵω2 )L(R) such that κ is regular in L(R) but κ is not a
cardinal in V .
Proof combines the determinacy theory of L(R), the Shelah p.c.f.
theory of V , and Woodin’s theory of the non-stationary ideal.
Woodin Covering Theorem:
Theorem. (Woodin) Same hypotheses as theorem. If A ⊆ λ <
ΘL(R) and |A| = ω1, then there is a B ∈ L(R), |B| = ω1 with
A ⊆ B.
A special case of this is the fact that every c.u.b. C ⊆ ω1 contains
a c.u.b. C1 ⊆ C with C1 ∈ L(R).
84
From Shelah’s p.c.f. theory we need the following.
Theorem (ZFC). Let A be a set of regular cardinals with |A| <
inf(A) and with cof(sup(A)) > ω. Then there is a c.u.b. C ⊆
sup(A) such that max p.c.f.(C +) = (sup(A))+, where C + = {κ+ : κ ∈
C}.
From the determinacy theory of L(R) we need the following fact.
Fact. For all α < ω1, δ 1α < ℵω1 .
We also need a certain partition property.
Let ~κ = {κα}α<ρ be an increasing, discontinuous sequence of regular cardinals.
Let θ~ = {θα}α<ρ be a ρ-sequence of ordinals with θα ≤ κα.
85
We consider block functions f from Σθα to sup κα.
~
We say ~κ → (~κ)θ if for any partition P of the block functions f
~ =
into two pieces, there is a blockwise c.u.b. homogeneous set H
{Hα}α<ρ for P (for functions blockwise of the correct type).
Let R ⊆ ℵω1 be the set of cardinals of the form δ 1α+1 for limit α.
ω
For κ ∈ R, there is pointclass Σ0 closed under ∃ω , ∧, ∨ω and
ω
scale(Σ0) such that if Π1 = ∀ω Σ0, then:
ω
Π1 is closed under ∀ω , ∩ω , ∪ω , scale(Π1), and δ 1α+1 = o(Π1).
Theorem (AD + DC). Let ~κ = {κα}α<ω1 ⊆ R. Then for all
θ < ω1 we have ~κ → (~κ)θ .
In fact, theorem holds for any ω1 sequence of pointclasses resembling Π11.
86
In special case ~κ ⊆ R of theorem, proof is easy as in proof of weak
partition relation on ω1 (special case easier since we can uniformly
find universal sets for the Π1 classes).
Theorem (AD + DC).
QLet µ be any measure on ω1. Let ~κ =
{κα}α<ω1 ⊆ R. Then κα/µ is regular.
Q
Proof. Let δ = κα/µ. Suppose π : λ → δ is cofinal, where λ < δ.
Consider the partition of block functions given by P([f ]µ, [g]µ) = 1
iff there is an element of ran(π) between [f ]µ and [g]µ. The homogeneous side must be the 1 side. Let S be block homogeneous
for P. For [f ]µ < δ, define [f 0] by f 0(α) = f (α)th element of S.
Then A = {[f 0]µ : [f ]µ < δ} has order-type δ and between any two
elements of A is an element of ran(π).
87
Proof of Theorem:
From the p.c.f. theory, let C ⊆ ℵω1 be a c.u.b. subset of the limit
Suslin cardinals such that max pcf(C +) = ℵω1+1. W.l.o.g. may
assume C ∈ L(R).
Let µ be the normal measure on ω1 (in L(R)), and let U (in V )
be an ultrafilter on ω1 extending µ.
Let f : ω1 → C + be increasing, f ∈ L(R), and f (α) > (δ 1α)+
almost everywhere.
.
Thus, λ = [f ]µ is regular in L(R). Assume λ is also regular in V ,
towards a contradiction. We also assume all elements of C + ⊆ R
are regular in V .
Q
.
Also, λ > ρ = ℵω1+1. In V , cof( f (α)/U) = ρ.
88
In V we define a cofinal map π : ρ → λ as follows. In V , fix a
scale {fα}α<ρ (only need U unboundedness).
For α < ρ, let π(α) = ([g]µ)L(R), where g ∈ L(R) represents the
µ least equivalence class such that g ≥ fα almost everywhere w.r.t.
U. Such a function exists by Woodin’s theorem, and the definition
is well-defined.
To show π is cofinal, let β < λ. Let [g0]µ = β. Pick α such that
fα >U g0. By definition π(α) = [g]µ where g ∈ L(R) and g >U fα.
So, g >U fα >U g0. Since g, g0 ∈ L(R), we have g >µ g0. Hence,
π(α) = [g]µ > [g0]µ = β.